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15-251 Great Theoretical Ideas in Computer Science.

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1 15-251 Great Theoretical Ideas in Computer Science

2 Algebraic Structures: Group Theory Lecture 15 (October 12, 2010)

3 Number Theory Naturals closed under + a+b = b+a a+0 = 0+a Integers closed under + a+b = b+a a+0 = 0+a a+(-a) = 0 ZnZn closed under + n a+ n b = b+ n a a+ n 0 = 0+ n a a+ n (-a) = 0 (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c)

4 A+0 = 0+A a+0 = 0+a a+(-a) = 0 a+ n 0 = 0+ n a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto A+(-A) = 0 1/a may not exist ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (A+B)+C = A+(B+C) Matrices Integers ZnZn (a+ n b)+ n c = a+ n (b+ n c)

5 A+0 = 0+A a+0 = 0+a a+(-a) = 0 a+ n 0 = 0+ n a a+ n (-a) = 0 closed under * closed under * n closed under * (a+b)*c = a*c+b*c ditto A+(-A) = 0 1/a exists if a  0 ditto Number Theory closed under + A+B = B+A closed under + a+b = b+a closed under + n a+ n b = b+ n a (a+b)+c = a+(b+c) (a+ n b)+ n c = a+ n (b+ n c) (A+B)+C = A+(B+C) Invertible Matrices Rationals Z n (n prime)

6 Abstraction: Abstract away the inessential features of a problem =

7 Today we are going to study the abstract properties of binary operations

8 Rotating a Square in Space Imagine we can pick up the square, rotate it in any way we want, and then put it back on the white frame

9 In how many different ways can we put the square back on the frame? R 90 R 180 R 270 R0R0 F|F| F—F— FF We will now study these 8 motions, called symmetries of the square

10 Symmetries of the Square Y SQ = { R 0, R 90, R 180, R 270, F |, F —, F, F }

11 Composition Define the operation “  ” to mean “first do one symmetry, and then do the next” For example, R 90  R 180 Question: if a,b  Y SQ, does a  b  Y SQ ? Yes! means “first rotate 90˚ clockwise and then 180˚” = R 270 F |  R 90 means “first flip horizontally and then rotate 90˚” = F

12 R 90 R 180 R 270 R0R0 F|F| F—F— FF R0R0 R 90 R 180 R 270 F|F| F—F— F F R0R0 R 90 R 180 R 270 F|F| F—F— FF R 90 R 180 R 270 F|F| F—F— F F R 180 R 270 R0R0 R0R0 R 90 R0R0 R 180 FFF|F| F—F— F—F— F|F| FF FFF—F— F|F| FF—F— F FF|F| F F—F— FF|F| F|F| FF—F— R0R0 R0R0 R0R0 R0R0 R 90 R 270 R 180 R 270 R 90 R 270 R 90 R 180 R 90 R 270 R 180

13 How many symmetries for n-sided body?2n R 0, R 1, R 2, …, R n-1 F 0, F 1, F 2, …, F n-1 R i R j = R i+j R i F j = F j-i F j R i = F j+i F i F j = R j-i

14 Some Formalism If S is a set, S  S is: the set of all (ordered) pairs of elements of S S  S = { (a,b) | a  S and b  S } If S has n elements, how many elements does S  S have? n2n2 Formally,  is a function from Y SQ  Y SQ to Y SQ  : Y SQ  Y SQ → Y SQ As shorthand, we write  (a,b) as “a  b”

15 “  ” is called a binary operation on Y SQ Definition: A binary operation on a set S is a function  : S  S → S Example: The function f:    →  defined by is a binary operation on  f(x,y) = xy + y Binary Operations

16 Is the operation  on the set of symmetries of the square associative? A binary operation  on a set S is associative if: for all a,b,c  S, (a  b)  c = a  (b  c) Associativity Examples: Is f:    →  defined by f(x,y) = xy + y associative? (ab + b)c + c = a(bc + c) + (bc + c)?NO! YES!

17 A binary operation  on a set S is commutative if For all a,b  S, a  b = b  a Commutativity Is the operation  on the set of symmetries of the square commutative? NO! R 90  F | ≠ F |  R 90

18 R 0 is like a null motion Is this true:  a  Y SQ, a  R 0 = R 0  a = a? R 0 is called the identity of  on Y SQ In general, for any binary operation  on a set S, an element e  S such that for all a  S, e  a = a  e = a is called an identity of  on S Identities YES!

19 Inverses Definition: The inverse of an element a  Y SQ is an element b such that: a  b = b  a = R 0 Examples: R 90 inverse: R 270 R 180 inverse: R 180 F|F| inverse: F |

20 Every element in Y SQ has a unique inverse

21 R 90 R 180 R 270 R0R0 F|F| F—F— FF R0R0 R 90 R 180 R 270 F|F| F—F— F F R0R0 R 90 R 180 R 270 F|F| F—F— FF R 90 R 180 R 270 F|F| F—F— F F R 180 R 270 R0R0 R0R0 R 90 R0R0 R 180 FFF|F| F—F— F—F— F|F| FF FFF—F— F|F| FF—F— F FF|F| F F—F— FF|F| F|F| FF—F— R0R0 R0R0 R0R0 R0R0 R 90 R 270 R 180 R 270 R 90 R 270 R 90 R 180 R 90 R 270 R 180

22 3. (Inverses) For every a  S there is b  S such that: Groups A group G is a pair (S,  ), where S is a set and  is a binary operation on S such that: 1.  is associative 2. (Identity) There exists an element e  S such that: e  a = a  e = a, for all a  S a  b = b  a = e

23 Commutative or “Abelian” Groups remember, “commutative” means a  b = b  a for all a, b in S If G = (S,  ) and  is commutative, then G is called a commutative group

24 To check “group-ness” Given (S,  ) 1.Check “closure” for (S,  ) (i.e, for any a, b in S, check a  b also in S). 2.Check that associativity holds. 3.Check there is a identity 4.Check every element has an inverse

25 Some examples…

26 Examples Is ( ,+) a group? Is + associative on  ? YES! Is there an identity?YES: 0 Does every element have an inverse?NO! ( ,+) is NOT a group Is  closed under +? YES!

27 Examples Is (Z,+) a group? Is + associative on Z? YES! Is there an identity?YES: 0 Does every element have an inverse?YES! (Z,+) is a group Is Z closed under +? YES!

28 Examples Is (Odds,+) a group? (Odds,+) is NOT a group Is + associative on Odds? YES! Is there an identity?NO! Does every element have an inverse?YES! Is Odds closed under +? NO!

29 Examples Is (Y SQ,  ) a group? Is  associative on Y SQ ? YES! Is there an identity?YES: R 0 Does every element have an inverse?YES! (Y SQ,  ) is a group the “dihedral” group D 4 Is Y SQ closed under  ? YES!

30 Examples Is (Z n,+ n ) a group? Is + n associative on Z n ? YES! Is there an identity?YES: 0 Does every element have an inverse?YES! (Z n, + n ) is a group (Z n is the set of integers modulo n) Is Z n closed under + n ? YES!

31 Examples Is (Z n,* n ) a group? Is * n associative on Z n ? YES! Is there an identity?YES: 1 Does every element have an inverse?NO! (Z n, * n ) is NOT a group (Z n is the set of integers modulo n)

32 Examples Is (Z n *, * n ) a group? Is * n associative on Z n * ? YES! Is there an identity?YES: 1 Does every element have an inverse? YES! (Z n *, * n ) is a group (Z n * is the set of integers modulo n that are relatively prime to n)

33 (Z, *) (Q, *) the rationals (Q \ {0}, *) No inverses… Zero has no inverse… Yes

34 3. (Inverses) For every a  S there is b  S such that: Groups A group G is a pair (S,  ), where S is a set and  is a binary operation on S such that: 1.  is associative 2. (Identity) There exists an element e  S such that: e  a = a  e = a, for all a  S a  b = b  a = e

35 Some properties of groups…

36 Theorem: A group has at most one identity element Proof: Suppose e and f are both identities of G=(S,  ) Then f = e  f = e Identity Is Unique We denote this identity by “e”  exactly one identity

37 Theorem: Every element in a group has a unique inverse Proof: Inverses Are Unique Suppose b and c are both inverses of a Then b = b  e = b  (a  c) = (b  a)  c = c

38 Theorem: If a  b = a  c, then b = c Proof: Cancellation

39 Orders and generators

40 A group G=(S,  ) is finite if S is a finite set Define |G| = |S| to be the order of the group (i.e. the number of elements in the group) What is the group with the least number of elements? How many groups of order 2 are there? G = ({e},  ) where e  e = e e f ef e f f e Order of a group

41 e a b e a b

42 Generators A set T  S is said to generate the group G = (S,  ) if every element of S can be expressed as a finite “sum” of elements in T Question: Does {R 90 } generate Y SQ ? Question: Does {F |, R 90 } generate Y SQ ? An element g  S is called a generator of G=(S,  ) if the set {g} generates G Does Y SQ have a generator? NO! YES! NO!

43 Generators For (Z n,+) Any a  Z n such that GCD(a,n)=1 generates (Z n,+) Claim: If GCD(a,n) =1, then the numbers a, 2a, …, (n-1)a, na are all distinct modulo n Proof (by contradiction): Suppose xa = ya (mod n) for x,y  {1,…,n} and x ≠ y Then n | a(x-y) Since GCD(a,n) = 1, then n | (x-y), which cannot happen

44 If G = (S,  ), we use a t denote (a  a  …  a) t times If G = (Z n, +), this means “ a t ” denotes (a + a + … + a) Order of an element If G = (Z n *, *), “ a t ” now denotes a t mod n Please be careful when using notation “ a t ” ! Warning: Potential Confusion = t*a mod n t times

45 If G = (S,  ), we use a t denote (a  a  …  a) t times Definition: The order of an element a of G is the smallest positive integer n such that a n = e The order of an element can be infinite! Example: The order of 1 in the group (Z,+) is infinite What is the order of F | in Y SQ ?2 What is the order of R 90 in Y SQ ?4 Order of an element

46 Remember order of a group G = size of the group G order of an element g in group G = (smallest n>0 s.t. g n = e)

47 Theorem: If G is a finite group, then for all g in G, order(g) is finite. Orders Proof: Consider g, g  g, g  g  g = g 3, g 4, … Since G is finite, g j = g k for some j < k  g j = g j  g k-j  e = g k-j Multiplying both sides by (g j ) -1

48 Remember order of a group G = size of the group G order of an element g = (smallest n>0 s.t. g n = e) g is a generator of group G if order(g) = order(G)

49 What is order(Z n, + n )? Orders n For x in (Z n, + n ), what is order(x)? order(x) = n/GCD(x,n)

50 Orders order(Z n *, * n )? Á (n) For x in (Z n, * n ), what is order(x)? At most Á (n) (Euler’s theorem)

51 Theorem: Let x be an element of G. The order of x divides the order of G Orders Corollary: If p is prime, a p-1 = 1 (mod p) (remember, this is Fermat’s Little Theorem) What group did we use for the corollary? G = (Z p *, *), order(G) = p-1

52 Groups and Subgroups

53 Subgroups Suppose G = (S,  ) is a group. If T µ S, and if H = (T,  ) is also a group, then H is called a subgroup of G.

54 Examples (Z, +) is a group and (Evens, +) is a subgroup. In fact, (Multiples of k, +) is also a subgroup. Is (Odds, +) a subgroup of (Z,+) ? No! (Odds,+) is not even a group!

55 Examples (Z n, + n ) is a group and if k | n, Is ({0, k, 2k, 3k, …, (n/k-1)k}, + n ) subgroup of (Z n,+ n ) ? Is (Z k, + k ) a subgroup of (Z n, + n )? Is (Z k, + n ) a subgroup of (Z n, + n )? No! it doesn’t even have the same operation No! (Z k, + n ) is not a group! (not closed) Only if k is a divisor of n.

56 Subgroup facts (identity) If e is the identity in G = (S,  ), what is the identity in H = (T,  )? e Proof: Clearly, e satisfies But we saw there is a unique such element in any group. e  a = a  e = a for all a in T.

57 Subgroup facts (inverse) If b is a’s inverse in G = (S,  ), what is a’s inverse in H = (T,  )? b Proof: let c be a’s inverse in H. Then c  a = e  c  a  b = e  b  c  e = b  c = b

58 Lagrange’s Theorem Theorem: If G is a finite group, and H is a subgroup then the order of H divides the order of G. In symbols, |H| divides |G|. Corollary: If x in G, then order(x) divides |G|. Proof of Corollary: Consider the set T x = (x, x 2 = x  x, x 3, …) H = (T x,  ) is a group. (check!) Hence it is a subgroup of G = (S,  ). Order(H) = order(x). (check!)

59 Lagrange’s Theorem Theorem: If G is a finite group, and H is a subgroup then the order of H divides the order of G. Curious (and super-useful) corollary: If you can show that H is a subgroup of G and H  G then |H| is at most ½ |G|

60 “Right” way of looking at primality testing Fermat: if n prime, then a n-1 = 1 (mod n) for all 0 < a < n. Suppose the converse was also true: “if n composite, then exists g with 0 < g < n. g n-1 != 1 (mod n)” Then consider “bad elements” for this n: elements b such that b n-1 = 1 (mod n) Bad elements form a subgroup of Z n Picking random element, it is good with probability ½.  |Bad| < ½ n Sadly, converse not true. Fixing that gives Miller-Rabin.

61 Symmetries of the Square Compositions Groups Binary Operation Identity and Inverses Basic Facts: Inverses Are Unique Generators Order of element, group Subgroups Lagrange’s theorem Here’s What You Need to Know…


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